1. Concept of Modeling Model -- The representation of an object or a system Modeling -- The creation and manipulation of an object or a system representation Model -- The representation of an object or a system Modeling -- The creation and manipulation of an object or a system representation

2. Concept of Modeling & Rendering Types of models -- Graphical model (geometric model) using geometric entities to describe the object, e.g., lines, polygons, curves, etc. -- Descriptive model mathematical or conceptual representation e.g., equations and attributes description. Types of models -- Graphical model (geometric model) using geometric entities to describe the object, e.g., lines, polygons, curves, etc. -- Descriptive model mathematical or conceptual representation e.g., equations and attributes description.

3. Model representation and modeling Line / Curve / Polygon / Surface Constructive solid-geometry and Hierarchical structure Hierarchical modeling with structures A hierarchical model can be created with structures by nesting the structures into one another to form a tree organization. Line / Curve / Polygon / Surface Constructive solid-geometry and Hierarchical structure Hierarchical modeling with structures A hierarchical model can be created with structures by nesting the structures into one another to form a tree organization.

4. Model representation and modeling Hierarchical modeling – from local coordinates to world coordinates (Modeling transformation) The graphical model is composed by a number of individual models.  transform individual local coordinates (model coordinates) to world coordinates  The graphical objects are constructed by transforming (e.g., rotating, scaling, translating, etc) the local individual objects from local coordinates to the world coordinates (this process is referred to as modeling transformation) Hierarchical modeling – from local coordinates to world coordinates (Modeling transformation) The graphical model is composed by a number of individual models.  transform individual local coordinates (model coordinates) to world coordinates  The graphical objects are constructed by transforming (e.g., rotating, scaling, translating, etc) the local individual objects from local coordinates to the world coordinates (this process is referred to as modeling transformation) + + Local coordinate systemworld coordinate system P mc * M = P wc

5. Aspects in 3D object rendering Projection Parallel projection and perspective projection Depth Cueing In order to reduce the ambiguity of 3D objects in 2D display, the intensity of objects rendered varies based on the distance (depth) from the viewing position Visible line and surface detection - hidden line/surface removal - non-visible lines displayed by dashed lines Surface rendering Setting the surface intensity of objects based on the lighting conditions and the surface characteristics. Projection Parallel projection and perspective projection Depth Cueing In order to reduce the ambiguity of 3D objects in 2D display, the intensity of objects rendered varies based on the distance (depth) from the viewing position Visible line and surface detection - hidden line/surface removal - non-visible lines displayed by dashed lines Surface rendering Setting the surface intensity of objects based on the lighting conditions and the surface characteristics.

6. 3D Object Representation Boundary representation -- surface representation e.g., polygon facets spline surfaces, which can be converted to polygon mesh by the tesselation Solid representation (space-partitioning) -- represent the object by a set of small solids (e.g., cubes), so that the interior property is described. (e.g., octree structure). Boundary representation -- surface representation e.g., polygon facets spline surfaces, which can be converted to polygon mesh by the tesselation Solid representation (space-partitioning) -- represent the object by a set of small solids (e.g., cubes), so that the interior property is described. (e.g., octree structure).

7. Surface representation Polygon data -- Include geometric data and attribute data (1) Geometric data: -- spatial information Polygon list  Edge list  Vertex list (2) Attribute data: -- surface property, e.g., transparency, texture property, reflectance property Polygon data -- Include geometric data and attribute data (1) Geometric data: -- spatial information Polygon list  Edge list  Vertex list (2) Attribute data: -- surface property, e.g., transparency, texture property, reflectance property

8. Polygon Data Geometric data (vertex-edge-polygon format) Vertex list Edge list Polygon structure Vertices Edges Polygons { { { V1: x1, y1, z1; e1: V1, V2; P1: e1, e2, e3; V2: x2, y2, z2;e2: V2, V3; P2: e2, e4, e5; :::::::::::::::::::::::::::::: :::::::::::::::::::: Vn: xn, yn, zn em: Vi, Vj; Pk: Ei, Ej, Ek; } } } Geometric data (vertex-edge-polygon format) Vertex list Edge list Polygon structure Vertices Edges Polygons { { { V1: x1, y1, z1; e1: V1, V2; P1: e1, e2, e3; V2: x2, y2, z2;e2: V2, V3; P2: e2, e4, e5; :::::::::::::::::::::::::::::: :::::::::::::::::::: Vn: xn, yn, zn em: Vi, Vj; Pk: Ei, Ej, Ek; } } } V1 V3 V2 V4 V5 e1 e2 e3 e4 e7 e6 e5 P1 P2 P3

9. Polygon Data Geometric data (vertex-polygon format) Vertex list Polygon structure Vertices Polygons { V1: x1, y1, z1; P1: V1, V2, V3; V2: x2, y2, z2; P2: V2, V3, V4; :::::::::::: :::::::::::::::::::: Vn: xn, yn, zn Pk: Vi, Vj, Vk; } } Geometric data (vertex-polygon format) Vertex list Polygon structure Vertices Polygons { V1: x1, y1, z1; P1: V1, V2, V3; V2: x2, y2, z2; P2: V2, V3, V4; :::::::::::: :::::::::::::::::::: Vn: xn, yn, zn Pk: Vi, Vj, Vk; } } V1 V3 V2 V4 V5 e1 e2 e3 e4 e7 e6 e5 P1 P2 P3

10. Curved surface Polygon mesh approximation Parametric representation e.g., sphere (r,  -- longitude angle,  -- latitude angle) x= r cos(  )cos(  ) -  /2 <=  <=  /2 y = r cos(  )sin(  ) -  <=  <=  z = r sin(  ) x^(2) + y^(2) + z^(2)= r^(2) Polygon mesh approximation Parametric representation e.g., sphere (r,  -- longitude angle,  -- latitude angle) x= r cos(  )cos(  ) -  /2 <=  <=  /2 y = r cos(  )sin(  ) -  <=  <=  z = r sin(  ) x^(2) + y^(2) + z^(2)= r^(2)   r y z x P(x,y,z)

11. Curved surface Polygon mesh approximation Parametric representation e.g., Ellipsoid -- extension of a spherical surface (  -- longitude angle,  -- latitude angle) x= r x cos(  )cos(  ) -  /2 <=  <=  /2 y = r y cos(  )sin(  ) -  <=  <=  z = r z sin(  ) (x/ r x ) ^(2) + (x/ r y ) ^(2) + (x/ r z ) ^(2)= 1 Polygon mesh approximation Parametric representation e.g., Ellipsoid -- extension of a spherical surface (  -- longitude angle,  -- latitude angle) x= r x cos(  )cos(  ) -  /2 <=  <=  /2 y = r y cos(  )sin(  ) -  <=  <=  z = r z sin(  ) (x/ r x ) ^(2) + (x/ r y ) ^(2) + (x/ r z ) ^(2)= 1   r y z x P(x,y,z)